Topological analysis of tetracyanobenzene metal–organic framework

Metal–organic frameworks (MOFs) are vital in modern material science, offering unique properties for gas storage, catalysis, and drug delivery due to their highly porous and customizable structures. Chemical graph theory emerges as a critical tool, providing a mathematical model to represent the molecular structure of these frameworks. Topological indices/molecular descriptors are mathematical formulations applied to molecular models, enabling the analysis of physicochemical properties and circumventing costly lab experiments. These descriptors are crucial for quantitative structure-property and structure-activity relationship studies in mathematical chemistry. In this paper, we study the different molecular descriptors of tetracyanobenzene metal–organic framework. We also give numerical comparison of computed molecular descriptors.

Moreover, MOFs has various applications in sensing and detection.Their systems can be designed to change within the presence of precise molecules, making them beneficial in detecting pollution, pollutants, or even biomarkers for sicknesses.In electrochemical power garage, MOFs are being investigated to be used in batteries and supercapacitors.Their structure can contribute to the efficiency and ability of those electricity storage devices 16 .
Chemical structures are every so often considered as complex and vast.They can be challenging to analyze in their natural forms.However, mathematical analysis, particularly graph theory, has been instrumental in simplifying and interpreting these structures in different scientific fields.In graph theory, atoms are represented as vertices and the connections between them as edges, transforming complex chemical or molecular structures into more comprehensible forms.
A topological index is used in chemical graph theory to describe a molecular structure.A topological descriptor reflects important structural characteristics of molecules.Most importantly, it is a structural invariant, meaning its value does not change regardless of how the chemical graph is labeled or depicted.These descriptors have found extensive use in correlating and predicting various chemical properties 17,18 .Topological indices are increasingly used in the development of molecules with pharmacological benefits.See references [19][20][21][22] for further information.
Gutman and Trinajstc introduced the degree-based topological descriptors, known as the first and second Zagreb indices 23 .Since then, various such descriptors have been defined.In 1975, Randić proposed the Randić index R −1/2 24 .This concept was later expanded by Bollobás and Erdős 25 , who generalized the Randić index for any real number α .Further contributions include the atom bond connectivity index by Estrada et al. 26 , and the geometric arithmetic index GA, introduced by Vukicevic et al. 27 .In 2008, Doslic defined the first and second Zagreb coindices 28 .Additionally, Zhou and Trinajsti' c presented the general sum-connectivity index in 29 , refining their earlier sum-connectivity index described in 30 .For recent work, we refer to see [31][32][33][34] .
Consider a graph χ with a vertex set V (χ ) comprising n χ vertices and an edge set E(χ) containing m χ edges.The degree of a vertex u is the count of edges incident to it.The edges of χ can be categorized based on the degrees of their end vertices, denoted as ( u , v ) , where u and v are the degrees of the end vertices u and v, respectively.
Introduced by Gutman and Trinajstic in 1972 23 , and Das and Gutman 35 the first and second Zagreb indices respectively of χ are given by: In 2008, Doślić defined the first and second Zagreb coindices 36  Ghorbani and Azimi in 2012 39 , and Ranjini et al. in 2013 40 , defined the first and second multiple Zagreb indices and the redefined first, second, and third Zagreb indices, respectively: (1) Lastly, the generalized Zagreb index, as defined by Azari and Iranmanesh 43 : An essential aspect of this study is to address the problem of predicting the physical properties of TCNB MOFs through topological analysis.This analysis is pivotal in understanding the fundamental properties of these frameworks and plays a crucial role in the design and synthesis of similar compounds.By leveraging the calculated topological indices, we aim to create a more efficient pathway for synthesizing MOFs with desired properties.Additionally, this study enriches scientific databases with detailed topological data, facilitating further research in the field.The topological indices calculated in this study are instrumental in predicting various physical properties of TCNB MOFs, such as thermal stability, porosity, and electrical conductivity.These indices serve as a quantitative measure that can guide the synthesis of new MOFs with tailored properties.Furthermore, including these topological data in scientific databases enhances the scope of computational research, offering a valuable resource for researchers exploring the vast potential of MOFs.

Tetracyanobenzene metal-organic framework (TCNB MOFs)
Tetracyanobenzene metal-organic framework (TCNB MOFs) is a type of porous material, synthesized by reacting a metal salt with TCNB in a solvent, where metal ions coordinate with the cyano groups on TCNB molecules to form an interconnected network.Characterized by high porosity, thermal stability, and electrical conductivity, these MOFs are versatile in applications like gas storage, catalysis, and energy storage.The TCNB framework, a two-dimensional network involving transition metal ions (TM) like Ti, V, Cr, Fe, Co, Ni, Cu, and Zn with tetracyanobenzene, exhibits a unique structure that varies from metallic to half-metallic.Metals such as Ni, Fe, Zn, and Co result in fully metallic structures, while Ti, V, Cr, and Mn create half-metallic MOFs, showing a gap in one spin direction.The arrangement in the TCNB network, especially with Ti, V, Cr, and Co, displays a screening effect, where the properties of these metals are significantly influenced by spin-polarized electrons from the surrounding organic ligands, unlike in TM − Pc .This effect is absent in Ni − TCNB and Zn − TCNB structures [44][45][46] .

Results and discussion
The chemical structure of TCNB(m, n), depicted in Fig. 1, exhibits a horizontal expansion of m ≥ 1 and a vertical expansion of n ≥ 1 .This structure comprises a total of 25m + 25n + 32mn + 17 atoms and 32m + 32n + 44mn + 20 bonds.It features four distinct atom types, categorized by degrees ranging from 1 to 4, with further specifics provided in Table 1.Additionally, the structure includes five different bond types, classified based on the degrees of both end atoms, as elaborated in Table 2.
First Zagreb Index of TCNB(m, n) We will compute the first Zagreb index of TCNB(m, n), by using the formula: Tetracyanobenzene ( TCNB(m, n) ) structure consist of five types of edge partitions, given in Table 2.By using these values, we have

Frequency
Set of Vertices HM(TCNB(m, n)) The first, second, and third redefined Zagreb indices of TCNB(m, n)

The Reduced Second Zagreb index of TCNB(m, n)
The Third Zagreb index of TCNB(m, n) Vol.:(0123456789)

Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.

Comparative discussion and conclusion
In this study, we comprehensively analyzed various degree and neighborhood-based topological indices of the tetracyanobenzene transition metal organic network, TCNB(m, n) .This investigation, focusing on analytical results, highlighted the intricate relationships between these indices and the structural properties of the TCNB network, thereby revealing their potential applications in diverse scientific and industrial fields.
The correlations observed between the topological indices and the physical properties of TCNB MOFs, such as thermal stability, porosity, and electrical conductivity, underscore the utility of topological analysis in predicting MOF behavior and properties.This is pivotal for synthesizing new materials with desired characteristics and contributes significantly to the field of material science, particularly in designing MOFs for specific applications like gas storage, catalysis, or drug delivery.Employing mathematical methods, especially those from chemical graph theory, has allowed for a more efficient and accurate prediction of MOFs' physical properties.By simplifying complex molecular structures into understandable forms, these methods have proven effective in providing a quantitative framework for interpreting MOFs' structural attributes.The application of topological indices in MOF analysis presents promising research, opening possibilities for developing more efficient methods for predicting material properties and designing new materials with customized functionalities.Future research could extend this approach to other types of MOFs, further enhancing the predictive models for material properties and contributing to the advancement of computational research in this domain.

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In 2016, Gutman et al. established a relationship to measure Zagreb coindices for a graph χ with n χ vertices and m χ edges 37 : In 2013, Shirdel et al. introduced the Hyper-Zagreb index 38 : www.nature.com/scientificreports/Gutman et al. in 2014 introduced the reduced second Zagreb index 41 : In 2010, Ghorbani and Hosseinzadeh introduced a third version of the Zagreb index 42 :

Table 3 .
Comparison of Zagreb first, second and third indices.

Table 8 .
Comparison of Generalized Zagreb indices for different {r, s}.